Classifying Spaces with Virtually Cyclic Stabilisers for Certain Infinite Cyclic Extensions

Let G be an infinite cyclic extension, 1 -> B -> G -> Z -> 1, of a group B where the action of Z on the set of conjugacy classes of non-trivial elements of B is free. This class of groups includes certain ascending HNN-extensions with abelian or free base groups, certain wreath products by Z and the soluble Baumslag-Solitar groups BS(1,m) with |m|> 1. We construct a model for Evc(G), the classifying space of G for the family of virtually cyclic subgroups of G, and give bounds for the minimum dimension of Evc(G). We construct a 2-dimensional model for Evc(G) where G is a soluble Baumslag-Solitar BS(1,m) group with |m|>1 and we show that this model for Evc(G) is of minimal dimension.Comment: Improved construction in Section 4 which gives subsequently better estimates for the dimensions in Section 6. 15 page

Brown's criterion in Bredon homology

We translate Brown's criterion for homological finiteness properties to the setting of Bredon homology.Comment: 10 page

On the classifying space for the family of virtually cyclic subgroups for elementary amenable groups

We show that elementary amenable groups, which have a bound on the orders of their finite subgroups, admit a finite dimensional model for the classifying space with virtually cyclic isotropy.Comment: 15 pages; revised versio

The Brin-Thompson groups sV are of type F_\infty

We prove that the Brin-Thompson groups sV, also called higher dimensional Thompson's groups, are of type F_\infty for all natural numbers s. This result was previously shown for s up to 3, by considering the action of sV on a naturally associated space. Our key step is to retract this space to a subspace sX which is easier to analyze.Comment: Final version, in Pacific J. Math., 10 pages, 4 figure

A note on the Mittag–Leffler condition for Bredon-modules

In this note we show the Bredon-analogue of a result by Emmanouil and Talelli, which gives a criterion when the homological and cohomological dimensions of a countable group $G$ agree. We also present some applications to groups of Bredon-homological dimension $1$.Comment: 10 page

Analysis of BAC end sequences in oak, a keystone forest tree species, providing insight into the composition of its genome

<p>Abstract</p> <p>Background</p> <p>One of the key goals of oak genomics research is to identify genes of adaptive significance. This information may help to improve the conservation of adaptive genetic variation and the management of forests to increase their health and productivity. Deep-coverage large-insert genomic libraries are a crucial tool for attaining this objective. We report herein the construction of a BAC library for <it>Quercus robur</it>, its characterization and an analysis of BAC end sequences.</p> <p>Results</p> <p>The <it>Eco</it>RI library generated consisted of 92,160 clones, 7% of which had no insert. Levels of chloroplast and mitochondrial contamination were below 3% and 1%, respectively. Mean clone insert size was estimated at 135 kb. The library represents 12 haploid genome equivalents and, the likelihood of finding a particular oak sequence of interest is greater than 99%. Genome coverage was confirmed by PCR screening of the library with 60 unique genetic loci sampled from the genetic linkage map. In total, about 20,000 high-quality BAC end sequences (BESs) were generated by sequencing 15,000 clones. Roughly 5.88% of the combined BAC end sequence length corresponded to known retroelements while <it>ab initio </it>repeat detection methods identified 41 additional repeats. Collectively, characterized and novel repeats account for roughly 8.94% of the genome. Further analysis of the BESs revealed 1,823 putative genes suggesting at least 29,340 genes in the oak genome. BESs were aligned with the genome sequences of <it>Arabidopsis thaliana</it>, <it>Vitis vinifera </it>and <it>Populus trichocarpa</it>. One putative collinear microsyntenic region encoding an alcohol acyl transferase protein was observed between oak and chromosome 2 of <it>V. vinifera.</it></p> <p>Conclusions</p> <p>This BAC library provides a new resource for genomic studies, including SSR marker development, physical mapping, comparative genomics and genome sequencing. BES analysis provided insight into the structure of the oak genome. These sequences will be used in the assembly of a future genome sequence for oak.</p

An Eilenberg-Ganea phenomenon for actions with virtually cyclic stabilizers

In dimension 3 and above, Bredon cohomology gives an acurate purely algebraic description of the minimal dimension of the classifying space for actions of a group with stabilisers in any given family of subgroups. For some Coxeter groups and the family of virtually cyclic subgroups we show that the Bredon cohomological dimension is 2 while the Bredon geometric dimension is 3

On Bredon (co-)homological dimensions of groups

The objects of interest in this thesis are classifying spaces EFG for discrete groups G with stabilisers in a given family F of subgroups of G. The main focus of this thesis lies in the family Fvc(G) of virtually cyclic subgroups of G. A classifying space for this specific family is denoted by EG. It has a prominent appearance in the Farrell–Jones Conjecture. Understanding the finiteness properties of EG is important for solving the conjecture. This thesis aims to contribute to answering the following question for a group G: what is the minimal dimension a model for EG can have? One way to attack this question is using methods in homological algebra. The natural choice for a cohomology theory to study G-CW-complexes with stabilisers in a given family F is known as Bredon cohomology. It is the study of cohomology in the category of OFG-modules. This category relates to models for EFG in the same way as the category of G-modules relates to the study of universal covers of Eilenberg–Mac Lane spaces K(G; 1). In this thesis we study Bredon (co-)homological dimensions of groups. A major part of this thesis is devoted to collect existing homological machinery needed to study these dimensions for arbitrary families F. We contribute to this collection. After this we turn our attention to the specific case of F = Fvc(G). We derive a geometric method for obtaining a lower bound for the Bredon (co-)homological dimension of a group G for a general family F, and subsequently show how to exploit this method in various cases for F = Fvc(G). Furthermore we construct model for EG in the case that G belongs to a certain class of infinite cyclic extensions of a group B and that a model for EB is known. We give bounds on the dimensions of these models. Moreover, we use this construction to give a concrete model for EG, where G is a soluble Baumslag–Solitar group. Using this model we are able to determine the exact Bredon (co-)homological dimensions of these groups. The thesis concludes with the study of groups G of low Bredon dimension for the family Fvc(G) and we give a classification of countable, torsion-free, soluble groups which admit a tree as a model for E